Having characterized the regret of OMD algorithms, we turn our attention to implementation issues.
The map to the dual space and back in
Eq. (REF ) and Eq. (REF ) (Lines 3 and 5 in Algorithm REF ), have the following expression [1]}, respectively:
\(& \hat{x}_{t,i} = \left(\nabla \Phi (\mathbf {x}_t)\right)_i = \text{sign} (x_{t,i})\frac{|x_{t,i}|^{q-1}}{\left\Vert \mathbf {x}_t\right\Vert ^{q-2}_q}, \quad \text{for all }~i \in \mathcal {N}, \\& {y}_{t+1,i} = \left( \left(\nabla \Phi \right)^{-1}(\hat{\mathbf {y}}_{t+1}) \right)_i = \text{sign} (\hat{y}_{t+1,i}) \frac{|\hat{y}_{t+1,i}|^{p-1}}{\left\Vert \hat{\mathbf {y}}_{t+1}\right\Vert ^{p-2}_p}, \quad \text{for all }~i \in \mathcal {N}.\) 
